Efficient way to remove equation number in a begin{align} environment aside from begin{split}?

I’m learning LaTeX and writing code on overleaf.

I am particular about the alignment of equal (=) signs and after banging my head for a few hours, it seems best to keep all equations in a single align environment and then repeatedly use intertext{} for discussions.

However, I came across a segment in my practice reference that has a number of equations in a row unnumbered. My solution was to write nonumberat the end of each equation.

#1. Building a new begin{align*} inside of begin{align} gives me errors.

#2. Adding a begin{split} works until the last line which can be corrected with a single nonumber, but all equations are now left justified.

#3. If I enclose everything with a begin{center} the intertext{} are also centered. This is with a setlengthparindent{0pt} preamble. begin{centered} does no visual change to #2

Is there a better way? I would prefer to keep only the equations centered.

x
Here is the compiled output for #1:

x
Here’s the compiled output for #2:

x
Here’s the compiled output for #3:

x
Here’s my code for #1, the nonumber statements are in the last 5 lines:
(I’m sorry it’s a complicated mess)

begin{align} 
    psi_mathbf{k}(r) & = sum_{IJ}  mathrm{e}^{imathbf{k}cdot mathbf{R}_{IJ}}  phi_o{(mathbf{r} - mathbf{R}_{IJ}})
    \
    intertext{We add a translation of an arbitrary vector (mathbf{R'}) and obtain}
    psi_mathbf{k}(r) & = sum_{IJ}  mathrm{e}^{imathbf{k}cdot mathbf{R}_{IJ}}  phi_o{(mathbf{r + R' - R}_{IJ}}) \
                       & = sum_{IJ}  mathrm{e}^{imathbf{k}cdot mathbf{R}_{IJ}}  phi_o{(mathbf{r - (R_mathit{IJ} - R')}})
    \
    intertext{Now let us define:}
    Tilde{mathbf{R}}_{IJ} & = mathbf{R_mathit{IJ}-R'}
%end{align}
%%
%begin{align}
    intertext{Then, since the summation in (2) is over an infinite number of pairs ((I, J)), we can rewrite it as}
    psi_mathbf{k}mathbf{(r+R')} & = sum_{IJ}mathrm{e}^{imathbf{k} cdot (mathbf{Tilde{R}}_{IJ}+R')}  phi_o(mathbf{r - Tilde{R})}_{IJ} nonumber\
                                   & = sum_{IJ}mathrm{e}^{imathbf{k cdot R'}} cdot mathrm{e}^{imathbf{k}mathbf{Tilde{R}}_{IJ}}   phi_o(mathbf{r - Tilde{R})}_{IJ} nonumber\
                                   & = mathrm{e}^{imathbf{k cdot R'}} sum_{IJ} mathrm{e}^{imathbf{k}mathbf{Tilde{R}}_{IJ}}   phi_o(mathbf{r - Tilde{R})}_{IJ} nonumber\
                                   & = mathrm{e}^{imathbf{k cdot R'}} psi_mathbf{k}mathbf{(r)} nonumber
end{align}

Here is the code for #2:

begin{align} 
    psi_mathbf{k}(r) & = sum_{IJ}  mathrm{e}^{imathbf{k}cdot mathbf{R}_{IJ}}  phi_o{(mathbf{r} - mathbf{R}_{IJ}})
    \
    intertext{We add a translation of an arbitrary vector (mathbf{R'}) and obtain}
    psi_mathbf{k}(r) & = sum_{IJ}  mathrm{e}^{imathbf{k}cdot mathbf{R}_{IJ}}  phi_o{(mathbf{r + R' - R}_{IJ}}) \
                       & = sum_{I J}  mathrm{e}^{imathbf{k}cdot mathbf{R}_{IJ}}  phi_o{(mathbf{r - (R_mathit{IJ} - R')}})
    \
    intertext{Now let us define:}
    Tilde{mathbf{R}}_{IJ} & = mathbf{R_mathit{IJ}-R'} \
%end{align}
%%
    begin{split}
        intertext{Then, since the summation in (2) is over an infinite number of pairs ((I, J)), we can rewrite it as}
        psi_mathbf{k}mathbf{(r+R')} & = sum_{IJ}mathrm{e}^{imathbf{k} cdot (mathbf{Tilde{R}}_{IJ}+R')}  phi_o(mathbf{r - Tilde{R})}_{IJ} \
                                       & =sum_{IJ}mathrm{e}^{imathbf{k cdot R'}} cdot mathrm{e}^{imathbf{k}mathbf{Tilde{R}}_{IJ}}   phi_o(mathbf{r - Tilde{R})}_{IJ} \
                                       & = mathrm{e}^{imathbf{k cdot R'}} sum_{IJ} mathrm{e}^{imathbf{k}mathbf{Tilde{R}}_{IJ}}   phi_o(mathbf{r - Tilde{R})}_{IJ}\
                                       & = mathrm{e}^{imathbf{k cdot R'}} psi_mathbf{k}mathbf{(r)}
    end{split}
end{align}

Code for #3 is just code for #2 with begin{center} and end{center} at both ends

tilde{mathbf{R}}

documentclass{article}
usepackage{amsmath}

newcommand{eul}{mathrm{e}}
renewcommand{vec}[1]{mathbf{#1}}

begin{document}

begin{align} 
psi_{vec{k}}(vec{r})
  & = sum_{IJ} eul^{ivec{k}cdot vec{R}_{IJ}} 
                phi_o(vec{r} - vec{R}_{IJ})
\
intertext{We add a translation of an arbitrary vector (vec{R}') and obtain}
begin{split}
psi_{vec{k}}(vec{r})
  & = sum_{IJ} eul^{ivec{k}cdot vec{R}_{IJ}}
                phi_o(vec{r} + vec{R}' - vec{R}_{IJ})
\
  & = sum_{IJ} eul^{ivec{k}cdot vec{R}_{IJ}}
                phi_o(vec{r} - (vec{R}_{IJ} - vec{R}'))
end{split}
end{align}
Now let us define
begin{equation}
Tilde{vec{R}}_{IJ} = vec{R}_{IJ}-vec{R}'
end{equation}
Then, since the summation in (2) is over an infinite number of pairs ((I, J)), 
we can rewrite it as
begin{equation}
begin{split}
psi_{vec{k}}(vec{r}+vec{R}')
  & = sum_{IJ} eul^{ivec{k} cdot (tilde{vec{R}}_{IJ}+vec{R}')}
                phi_o(vec{r} - tilde{vec{R}})_{IJ} \
  & = sum_{IJ} eul^{ivec{k} cdot vec{R}'} cdot 
                eul^{ivec{k}cdottilde{vec{R}}_{IJ}}
                phi_o(vec{r} - tilde{vec{R}})_{IJ} \
  & = eul^{ivec{k} cdot vec{R}'}
      sum_{IJ} eul^{ivec{k}cdottilde{vec{R}}_{IJ}}
                phi_o(vec{r} - tilde{vec{R}})_{IJ} \
  & = eul^{ivec{k} cdot vec{R}'} psi_{vec{k}}(vec{r})
end{split}
end{equation}

end{document}